Modeling Charmonium-$\eta$ Decays of $J^{PC}=1^{--}$ Higher Charmonia

We propose a new model to create a light meson in the heavy quarkonium transition, which is inspired by the Nambu$-$Jona-Lasinio (NJL) model. Hadronic transitions of $J^{PC}=1^{--}$ higher charmonia with the emission of an $\eta$ meson are studied in the framework of the proposed model. The model shows its potential to reproduce the observed decay widths and make predictions for the unobserved channels. We present our predictions for the decay width of $\Psi \to J/\psi \eta$ and $\Psi \to h_{c}(1P)\eta$, where $\Psi$ are higher $S$ and $D$ wave vector charmonia, which provide useful references to search for higher charmonia and determine their properties in forthcoming experiments. The predicted branching fraction $\mathcal B(\psi(4415)\to h_{c}(1P)\eta)=4.62\times 10^{-4}$ is one order of magnitude smaller than the $J/\psi \eta$ channel. Estimates of partial decay width $\Gamma(Y \to J/\psi \eta)$ are given for $Y(4360)$, $Y(4390)$ and $Y(4660)$ by assuming them as $c\bar{c}$ bound states with quantum numbers $3 ^3D_1$, $3 ^3D_1$ and $5 ^3S_1$, respectively. Our results are in favor of these assignments for $Y(4360)$ and $Y(4660)$. The corresponding experimental data for these $Y$ states has large statistical errors which do not provide any constraint on the mixing angle if we introduce $S-D$ mixing. To identify $Y(4390)$, precise measurements on its hadronic branching fraction are required which are eagerly awaited from BESIII.